ALINA STANCU, Courant Institute of Mathematical Sciences, New York,
New York 10012, USA

Asymptotic behavior of a crystalline evolution

Motion by crystalline curvature is viewed as a typical example of
geometric evolution by a nonsmooth boundary energy. Assume that a
planar curve is endowed with an energy density defined on a finite set
of normal directions. It is natural then to consider the restricted
class of piecewise linear curves with just this ordered set of
normals. These curves do not have a motion by curvature in the
conventional geometric sense, but, following M. Gurtin and J. Taylor,
one can still define the so-called crystalline curvature flow which is
analogous to the motion by weighted curvature for smooth planar
curves.

We consider Gurtin's defintion, with no driving term, for closed
convex
curves, so that the inward normal velocity of each segment of an
admissible polygonal curve as above is inversely proportional to the
length of the segment, where the proportionality factor is only
required to be positive. Our results show that, if the curve has more
than four sides, it will shrink to a point while approaching the shape
of a homothetic solution to the flow. This implies the existence of
at
least one self similar solution for any flow associated to an energy
density defined on more than four unitary directions. The number of
homothetic solutions will be discussed based on the properties of the
energy density.